Match a with b so that a is congruent to b modulo 7.a = 23Answer 1a = 11Answer 2a = 17Answer 3

To determine which answer is congruent to $a$ modulo 7, we will perform the calculations for each option.

Step 1: Break Down the Problem

We need to find the values of $a = 23$, $a = 11$, $a = 17$, and determine the equivalences of these values modulo 7.

Step 2: Relevant Concepts

The concept of congruence modulo $n$ states that two numbers $x$ and $y$ are congruent modulo $n$ if $x \equiv y \mod n$. This can be expressed as: $x \mod n = y \mod n$

Step 3: Analysis and Detail

1. Calculate $23 \mod 7$: $23 \div 7 = 3 \quad (\text{quotient})$ $23 - (3 \times 7) = 23 - 21 = 2$ So, $23 \equiv 2 \mod 7$.

2. Calculate $11 \mod 7$: $11 \div 7 = 1 \quad (\text{quotient})$ $11 - (1 \times 7) = 11 - 7 = 4$ So, $11 \equiv 4 \mod 7$.

3. Calculate $17 \mod 7$: $17 \div 7 = 2 \quad (\text{quotient})$ $17 - (2 \times 7) = 17 - 14 = 3$ So, $17 \equiv 3 \mod 7$.

Step 4: Verify and Summarize

• $23 \equiv 2 \mod 7$
• $11 \equiv 4 \mod 7$
• $17 \equiv 3 \mod 7$

None of the answers provided match $23 \equiv 2 \mod 7$.

Final Answer

None of the options (11, 17) are congruent to $23$ modulo 7.